#8599749

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,8)(4,5)(6,7)(17,18,23,25)(19,24,28,20)(21,30,29,26)(22,27,31,32);(1,3,6,5)(2,4,7,8)(10,13)(14,16)(17,19)(18,24)(20,25)(21,31)(22,29)(23,28)(26,27)(30,32);(2,7)(4,8)(9,10,12,13)(11,14,15,16)(18,25)(19,28)(21,29)(27,32);(2,7)(3,5)(11,15)(14,16)(17,20)(18,19)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,29)(18,26,25,30)(19,22,28,31)(20,27,24,32);(1,5,6,3)(2,4,7,8)(9,11)(10,14)(12,15)(13,16)(17,22)(18,27)(19,29)(20,30)(21,28)(23,31)(24,26)(25,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506945);